$\begingroup$I haven't ran any searches yet, but I have a hunch the OEIS covers this in excruciating detail.$\endgroup$
– Mr. BrooksJun 20 '17 at 21:51

$\begingroup$@Mr.Brooks The only relevant sequence I found is the sequence of Concatenators: A121609, but I haven't found anything that is similar to delayed concatenators.$\endgroup$
– VepirJun 21 '17 at 9:38

$\begingroup$Well, maybe that's something that needs to be added to the OEIS. I've thought of several sequences but I look them up and I find they're already in the OEIS.$\endgroup$
– Mr. BrooksJun 23 '17 at 19:51

$\begingroup$is there an easy (read: algorithmically fast) way to tell if a pair $(p,q)$ does not have a concatenator?$\endgroup$
– TommyAug 31 '17 at 15:51

1 Answer
1

Note that $1,10,10^2,10^3,10^4$ and $10^5$ all have different remainders mod $7$. Thus, if neither $p$ nor $q$ is $7$, one of $p\|q,10p\|q,...,10^5p\|q$ is a multiple of $7$. So you can't get a delay of more than $5$ unless one of the primes is $7$.